Nonmonotone systems decomposable into monotone systems with negative feedback |
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Authors: | GA Enciso ED Sontag |
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Institution: | a Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA b Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA c Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA |
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Abstract: | Motivated by the work of Angeli and Sontag Monotone control systems, IEEE Trans. Automat. Control 48 (2003) 1684-1698] and Enciso and Sontag On the global attractivity of abstract dynamical systems satisfying a small gain hypothesis, with applications to biological delay systems, Discrete Continuous Dynamical Systems, to appear] in control theory, we show that certain finite and infinite dimensional semi-dynamical systems with “negative feedback” can be decomposed into a monotone “open-loop” system with “inputs” and a decreasing “output” function. The original system is reconstituted by “plugging the output into the input”. Employing a technique of Gouzé A criterion of global convergence to equilibrium for differential systems with an application to Lotka-Volterra systems, Rapport de Recherche 894, INRIA] and Cosner Comparison principles for systems that embed in cooperative systems, with applications to diffusive Lotka-Volterra models, Dynam. Cont., Discrete Impulsive Systems 3 (1997) 283-303] of imbedding the system into a larger symmetric monotone system, we are able to obtain information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence. |
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